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Random Zero-Sum Dynamic Games on Infinite Directed Graphs.

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  • 1CEREMADE, Paris Dauphine University, Paris, France.

Dynamic Games and Applications
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PubMed
Summary
This summary is machine-generated.

This study analyzes random two-player zero-sum dynamic games on infinite graphs. We found that game values converge exponentially for certain graphs and double-exponentially for infinite d-ary trees as game duration increases.

Keywords:
Directed graphsDynamic gamesRandom gamesZero-sum games

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Area of Science:

  • Game Theory
  • Probability Theory
  • Graph Theory

Background:

  • Two-player zero-sum dynamic games with perfect information are analyzed.
  • Games are played on infinite directed graphs with vertex-assigned payoffs.
  • Payoffs are distributed i.i.d. (independently and identically distributed) across vertices.

Purpose of the Study:

  • To investigate the convergence of game values in random dynamic games.
  • To analyze convergence rates on different classes of infinite directed graphs.
  • To understand the impact of graph structure and game duration on game values.

Main Methods:

  • Consideration of random two-player zero-sum dynamic games with perfect information.
  • Analysis on classes of infinite directed graphs, including acyclic graphs and the infinite d-ary tree.
  • Asymptotic analysis of game values as the duration of the game tends to infinity.

Main Results:

  • For acyclic directed graphs with bounded degree and sub-exponential expansion, game values converge almost surely to a constant at an exponential rate.
  • For the infinite d-ary tree, convergence of game values is shown to occur at a double-exponential rate.
  • The convergence rate is dominated by the expansion properties of the graph.

Conclusions:

  • The structure of the infinite directed graph significantly influences the convergence rate of game values.
  • Exponential convergence is observed in graphs with controlled expansion, while faster double-exponential convergence occurs in the infinite d-ary tree.
  • These findings provide insights into the asymptotic behavior of random dynamic games on complex graph structures.